For how to determine the transcription factor binding sites in the whole genome, I'd use ChIP with ChIP sequencing. I'm more than 100% positive that's the way my professor expects us to do it since he just talked about Monday.

As for the for the first question, I guess use a DNAse1 Foot-Printing Assay? A classmate of mine mentioned knockout mutations but, it's not asking what the gene does. So I don't see how that would be correct.

Actually you are wrong and your friend is right. The DNase foot-printing is good to recognize the particular binding place. I.e. answer to question 2. But to recognise, which genes are controled by this TF (without being interested in actual binding site), you need to change it's expression and see, what changes in transcription of other genes you get